Projection, Consistency, and George Boole

Transcription

1 Projection, Consistency, and George Boole John Hooker Carnegie Mellon University CP 2015, Cork, Ireland

2 Projection as a Unifying Concept Projection underlies both optimization and logical inference. Optimization is projection onto a cost variable. Logical inference is projection onto a subset of variables. These 3 concepts are linked in George Boole s work on probability logic. 2

3 Projection as a Unifying Concept Projection underlies both optimization and logical inference. Optimization is projection onto a cost variable. Logical inference is projection onto a subset of variables. These 3 concepts are linked in George Boole s work on probability logic. Consistency maintenance can likewise be seen as projection. Leads to a simple type of consistency based eplicitly on projection. 3

4 Probability Logic George Boole is best known for propositional logic and Boolean algebra. But he proposed a highly original approach to probability logic. 4

5 Logical Inference The problem: Given a set S of propositions Each with a given probability. And a proposition P that can be deduced from S What probability can be assigned to P? 5

6 Boole s Probability Logic In 1970s, Theodore Hailperin offered an interpretation of Boole s probability logic based on modern concept of linear programming. LP first clearly formulated in 1930s by Kantorovich. Hailperin (1976) 6

7 Boole s Probability Logic if if Clause 1 Eample then then 2 3 Probability Deduce probability range for 3

8 Boole s Probability Logic if if Clause 1 Eample then then 2 3 Probability Deduce probability range for 3 Interpret if-then statements as material conditionals

9 Boole s Probability Logic Clause 1 Eample 2 3 Probability Deduce probability range for 3 Interpret if-then statements as material conditionals

10 Boole s Probability Logic Clause 1 Eample 2 3 Probability Deduce probability range for 3 Linear programming model min/ ma p p p p p 000 = probability that ( 1, 2, 3 ) = (0,0,0) 10

11 Boole s Probability Logic Clause 1 Eample 2 3 Probability Deduce probability range for 3 Linear programming model min/ ma p p p p p 000 = probability that ( 1, 2, 3 ) = (0,0,0) Solution: 0 [0.1, 0.4] 11

12 Boole s Probability Logic This LP model was re-invented in AI community. Nilsson (1986) Column generation methods are now available. To deal with eponential number of variables. 12

13 Projection An LP can be solved by Fourier elimination The only known method in Boole s day This is a projection method. Fourier (1827) 13

14 Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b

15 Projection Eliminate variables we want to project out. To solve min/ ma y y ay, Ay b 0 0 project out all variables y j ecept y 0. To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j where c, d

16 Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d

17 Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d

18 Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d c d y c0 d0 c0 d0 18

19 Projection Projection as a common theme: Optimization: Project onto objective function variable Logical inference: Project onto propositional variables of interest Consistency maintenance:??? Look at logical inference net 19

20 Inference as Projection Project onto propositional variables of interest Suppose we wish to infer from these clauses everything we can about propositions 1, 2, 3 20

21 Inference as Projection Project onto propositional variables of interest Suppose we wish to infer from these clauses everything we can about propositions 1, 2, 3 We can deduce 1 3 This is a projection onto 1, 2, 3 21

22 Inference as Projection Resolution as a projection method Similar to Fourier elimination Actually, identical to Fourier elimination + rounding To project out j, eliminate it from pairs of clauses: C j D j C D Quine (1952,1955) JH (1992,2012) 22

23 Inference as Projection Resolution as a projection method Similar to Fourier elimination Actually, identical to Fourier elimination + rounding To project out j, eliminate it from pairs of clauses: C j D j C D This is too slow. Another approach is logic-based Benders decomposition 23

24 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem Benders cut from previous iteration 24

25 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,0) Resulting subproblem 25

26 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,0) Resulting subproblem Subproblem is infeasible. ( 1, 3 )=(0,0) creates infeasibility 26

27 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem 1 3 solution of master ( 1, 2, 3 ) = (0,1,0) Benders cut (nogood) Resulting subproblem Subproblem is infeasible. ( 1, 3 )=(0,0) creates infeasibility 27

28 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem 1 3 solution of master ( 1, 2, 3 ) = (0,1,1) Resulting subproblem 28

29 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem 1 3 solution of master ( 1, 2, 3 ) = (0,1,1) Resulting subproblem Subproblem is feasible 29

30 Inference as Projection Benders decomposition computes a projection Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Enumerative Benders cut Subproblem is feasible Resulting subproblem 30

31 Inference as Projection Benders decomposition computes a projection Logic-based Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Enumerative Benders cut Resulting subproblem JH and Yan (1995) JH (2012) Continue until master is infeasible. Black Benders cuts describe projection. 31

32 Inference as Projection Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2 first. 32

33 Inference as Projection Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2 first. Conflict clauses 33

34 Inference as Projection Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2 first. Conflict clauses Backtrack to 3 at feasible leaf nodes 34

35 Inference as Projection Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2 first. Conflict clauses containing 1, 2, 3 describe projection 35

36 Inference as Projection Projection methods similar to Fourier elimination For logical clauses Quine (1952,1955) For cardinality clauses JH (1988) For 0-1 linear inequalities JH (1992) For general integer linear inequalities Williams & JH (2015) 36

37 Consistency as Projection Domain consistency Project onto each individual variable j. 37

38 Consistency as Projection Domain consistency Project onto each individual variable j. Eample: Constraint set alldiff,, a, b a, b b, c 38

39 Consistency as Projection Domain consistency Project onto each individual variable j. Eample: Constraint set alldiff,, 3 a, b a, b b, c Solutions,, 3 ( a, b, c) ( bac,, ) 39

40 Consistency as Projection Domain consistency Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, 3 a, b a, b b, c,, 3 ( a, b, c) ( bac,, ) a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 40

41 Consistency as Projection Domain consistency Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, 3 a, b a, b b, c,, 3 ( a, b, c) ( bac,, ) This achieves domain consistency. a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 41

42 Consistency as Projection Domain consistency Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, 3 a, b a, b b, c,, 3 ( a, b, c) ( bac,, ) This achieves domain consistency. We will regard a projection as a constraint set. a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 42

43 Consistency as Projection k-consistency Can be defined: A constraint set S is k-consistent if: for every J {1,, n} with J = k 1, every assignment J = v J D j for which ( J, j ) does not violate S, and every variable j J, J = ( j j J) there is an assignment j = v j D j for which ( J, j ) = (v J,v j ) does not violate S. 43

44 Consistency as Projection k-consistency Can be defined: A constraint set S is k-consistent if: for every J {1,, n} with J = k 1, every assignment J = v J D j for which ( J, j ) does not violate S, and every variable j J, there is an assignment j = v j D j for which ( J, j ) = (v J,v j ) does not violate S. To achieve k-consistency: J = ( j j J) Project the constraints containing each set of k variables onto subsets of k 1 variables. 44

45 Consistency as Projection Consistency and backtracking: Strong k-consistency for entire constraint set avoids backtracking if the primal graph has width < k with respect to branching order. Freuder (1982) No point in achieving strong k-consistency for individual constraints if we propagate through domain store. Domain consistency has same effect. 45

46 J-Consistency A type of consistency more directly related to projection. Constraint set S is J-consistent if it contains the projection of S onto J. S is domain consistent if it is { j }-consistent for each j. Resolution and logic-based Benders achieve J-consistency for SAT. J = ( j j J) 46

47 J-Consistency J-consistency and backtracking: If we branch on variables 1, 2,, a natural strategy is to project out n, n1, until computational burden is ecessive. 47

48 J-Consistency J-consistency and backtracking: If we branch on variables 1, 2,, a natural strategy is to project out n, n1, until computational burden is ecessive. No point in achieving J-consistency for individual constraints if we propagate through a domain store. However, J-consistency can be useful if we propagate through a richer data structure such as decision diagrams which can be more effective as a propagation medium. Andersen, Hadžić JH, Tiedemann (2007) Bergman, Ciré, van Hoeve, JH (2014) 48

49 Propagating J-Consistency Eample: among (, ),{ c, d},1,2 ( c) ( d), a, b, c, d alldiff,,, 3 4 a, b c, d 3 4 Already domain consistent for individual constraints. If we branch on 1 first, must consider all 4 branches 1 = a,b,c,d 49

50 Eample: ( c) ( d), a, b, c, d Propagating J-Consistency among (, ),{ c, d},1,2 alldiff,,, 3 4 a, b c, d 3 4 a,b,d a,b,c,d Suppose we propagate through a relaed decision diagram of width 2 for these constraints c d 52 paths from top to bottom represent assignments to 1, 2, 3, 4 36 of these are the feasible assignments. a,b c,d 50

51 Eample: ( c) ( d), a, b, c, d Propagating J-Consistency among (, ),{ c, d},1,2 alldiff,,, 3 4 a, b c, d 3 4 a,b,d a,b,c,d 52 paths from top to bottom represent assignments to 1, 2, 3, 4 36 of these are the feasible assignments. Suppose we propagate through a relaed decision diagram of width 2 for these constraints a,b c,d c d Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 51

52 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. a,b,d c For each arc, indicate length of shortest path from top to that arc. a,b,c,d a,b c,d d Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 52

53 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. a,b,d 0,0,1 c 1 For each arc, indicate length of shortest path from top to that arc. a,b,c,d 0,0,1,1 a,b c,d d 2 Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 53

54 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. a,b,d 0,0,1 c 1 Remove arcs with label > 1 For each arc, indicate length of shortest path from top to that arc. a,b,c,d 0,0,1,1 a,b c,d d 2 Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 54

55 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. a,b,d 0,0,1 c 1 Remove arcs with label > 1 For each arc, indicate length of shortest path from top to that arc. a,b,c,d 0,0,1,1 a,b c,d Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 55

56 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. a,b,d 0,0,1 c 1 Remove arcs with label > 1 Clean up. For each arc, indicate length of shortest path from top to that arc. a,b,c,d 0,0,1,1 a,b c,d Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 56

57 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. Let the length of a path be number of arcs with labels in {c,d}. For each arc, indicate length of shortest path from top to that arc. a,b,d 0,0,1 a,b,c,d 0,0,1,1 a,b c,d Remove arcs with label > 1 Clean up. Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 57

58 Propagating J-Consistency Let s propagate the 2 nd atmost constraint in the projected alldiff through the relaed decision diagram. We need only branch on a,b,d rather than a,b,c,d a,b,d Remove arcs with label > 1 Clean up. a,b,c,d a,b c,d Projection of alldiff onto 1, 2 is alldiff, atmost (, ),{ a, b},1 atmost (, ),{c,d},1 58

59 Achieving J-consistency Constraint How hard to project? among sequence regular alldiff Easy and fast. More complicated but fast. Easy and basically same labor as domain consistency. Quite complicated but practical for small domains. 59

60 Projection of J-consistency for Among among (,, ), V, t, u where 1 among ( 1,, n 1), V, t, u onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise n 60

61 Eample Projection of J-consistency for Among among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u

62 Eample Projection of J-consistency for Among among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 62

63 Eample Projection of J-consistency for Among among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 63

64 Eample Projection of J-consistency for Among among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} 64

65 Eample Projection of J-consistency for Among among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} 65

66 J-consistency for Among Eample Projection of among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} among (),{ c, d},( t 4),min{ u 2,0} 66

67 J-consistency for Among Eample Projection of among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} among (),{ c, d},( t 4),min{ u 2,0} Feasible if and only if ( t 4) minu 2,0 67

68 J-Consistency for Sequence Projection is based on an integrality property. The coefficient matri of the inequality formulation has consecutive ones property. So projection of the conve hull of the feasible set is an integral polyhedron. Polyhedral projection therefore suffices. Straightforward (but tedious) application of Fourier elimination yields the projection. 68

69 J-Consistency for Sequence Projection is based on an integrality property. The coefficient matri of the inequality formulation has consecutive ones property. So projection of the conve hull of the feasible set is an integral polyhedron. Polyhedral projection therefore suffices. Straightforward (but tedious) application of Fourier elimination yields the projection. Projection onto any subset of variables is a generalized sequence constraint. Compleity of projecting out k is O(kq), where q = length of the overlapping sequences. 69

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71 J-Consistency for Sequence Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1,

72 J-Consistency for Sequence Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0,

73 J-Consistency for Sequence Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0, To project out 4, add constraints among ( ),{1},1,1 among (,, ),{1},1,2 among (, ),{1},0,1 among ( ),{1},0,

74 J-Consistency for Sequence Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0, To project out 4, add constraints among ( ),{1},1,1 among (,, ),{1},1,2 among (, ),{1},0,1 among ( ),{1},0, To project out 3, fi ( 1, 2 ) = (1,1) 74

75 J-Consistency for Regular Constraint Projection can be read from state transition graph. Compleity of projecting onto 1,, k for all k is O(nm 2 ), where n = number of variables, m = ma number of states per stage. 75

76 J-Consistency for Regular Constraint Projection can be read from state transition graph. Compleity of projecting onto 1,, k for all k is O(nm 2 ), where n = number of variables, m = ma number of states per stage. Shift scheduling eample Assign each worker to shift i {a,b,c} on each day i = 1,,7. Must work any given shift 2 or 3 days in a row. No direct transition between shifts a and c. Variable domains: D 1 = D 5 = {a,c}, D 2 = {a,b,c}, D 3 = D 6 = D 7 = {a,b}, D 4 = {b,c} 76

77 J-Consistency for Regular Constraint Deterministic finite automaton for this problem instance: = absorbing state Regular language epression: (((aa aaa)(bb bbb))* ((cc ccc)(bb bbb))*)*(c (aa aaa) (cc ccc)) 77

78 J-Consistency for Regular Constraint State transition graph for 7 stages Dashed lines lead to unreachable states. 78

79 J-Consistency for Regular Constraint State transition graph for 7 stages Dashed lines lead to unreachable states. Filtered domains 79

80 J-Consistency for Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. 80

81 J-Consistency for Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. 81

82 J-Consistency for Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. Resulting graph can be viewed as a constraint that describes the projection. Constraint is easily propagated through a relaed decision diagram. 82

83 J-Consistency for Alldiff Constraint Projection is inherently complicated. But it can simplify for small domains. The result is a disjunction of constraint sets, each of which contains an alldiff constraint and some atmost constraints. 83

84 J-Consistency for Alldiff Constraint Eample alldiff,,,, 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g

85 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, 85

86 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. 86

87 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. 87

88 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get a f g alldiff,,, atmost (,, ),{,, },

89 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get a f g alldiff,,, atmost (,, ),{,, },1 3 3 If 4 {a,f,g}, we get 4 = e, and we remove e from other domains. 89

90 J-Consistency for Alldiff Constraint Eample alldiff,,,, Projecting out 5, we get 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get a f g alldiff,,, atmost (,, ),{,, },1 3 3 If 4 {a,f,g}, we get 4 = e, and we remove e from other domains. So the projection is D1 a, b, c alldiff 1, 2, 3 D2 c, d atmost ( 1, 2, 3 ),{ a, f, g},1 D 3 d, f 90

91 J-Consistency for Alldiff Constraint Eample alldiff,,,, 3 4 5,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g Projecting out 3, we get simply alldiff, Projecting out 2, we get the original domain for 1 D1 a, b, c 91

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93 Bounds Consistency as Projection Bounds consistency Most naturally defined when domains can be embedded in the real numbers. Then we achieve bounds consistency by projecting the conve hull of the feasible set onto each j. Continuous J-consistency is achieved by projecting the conve hull onto J. 93

94 Bounds Consistency as Projection Cutting planes Projection onto J is defined by cutting planes that contain variables in J. Close relationship to integer programming.. 94

95 Bounds Consistency as Projection Usefulness of cutting planes They can be propagated through LP relaation. This can help bound the objective function as well as achieve consistency. They can reduce backtracking Even when LP relaation is not used. This has never been studied in integer programming! 95